Page 438 - Handbook Of Integral Equations

b

59. y(x)+ y(x – t)f t, y(t) dt = A sin λx.
a
A solution:
y(x)= p sin λx + q cos λx, (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:
b

p + (p cos λt + q sin λt)f t, p sin λt + q cos λt dt = A,
a
b (2)

q + (q cos λt – p sin λt)f t, p sin λt + q cos λt dt =0.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b

60. y(x)+ y(x – t)f t, y(t) dt = A cos λx.
a
A solution:
y(x)= p sin λx + q cos λx,
where p and q are roots of the following system of algebraic (or transcendental) equations:
b

p + (p cos λt + q sin λt)f t, p sin λt + q cos λt dt =0,
a
b

q + (q cos λt – p sin λt)f t, p sin λt + q cos λt dt = A.
a
b
µx
61. y(x)+ y(x – t)f t, y(t) dt = e (A sin λx + B cos λx).
a
A solution:
µx
y(x)= e (p sin λx + q cos λx), (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:
b

p + (p cos λt + q sin λt)e –µt f t, pe µt sin λt + qe µt cos λt dt = A,
a
b (2)

q + (q cos λt – p sin λt)e –µt f t, pe µt sin λt + qe µt cos λt dt = B.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b

62. y(x)+ y(x – t)f t, y(t) dt = g(x).
a
◦
1 .For g(x)= n A k exp(λ k x), the equation has a solution of the form
k=1
n

y(x)= B k exp(λ k x),
k=1
where the constants B k are determined from the nonlinear algebraic (or transcendental) system
%
B k + B k F k (B) – A k =0, k =1, ... , n,
n
b
%
%
B = {B 1 , ... , B n }, F k (B)= f t, B m exp(λ m t) exp(–λ k t) dt.
a
m=1
Different solutions of this system generate different solutions of the integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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